@article{RiberaBorrellQuerRichteretal.2023, author = {Ribera Borrell, Enric and Quer, Jannes and Richter, Lorenz and Sch{\"u}tte, Christof}, title = {Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics}, volume = {89}, journal = {SIAM Journal on Scientific Computing (SISC)}, number = {1}, doi = {10.1137/22M1503464}, year = {2023}, abstract = {Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings.}, language = {en} } @misc{RiberaBorrellQuerRichteretal.2021, author = {Ribera Borrell, Enric and Quer, Jannes and Richter, Lorenz and Sch{\"u}tte, Christof}, title = {Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics}, issn = {1438-0064}, year = {2021}, abstract = {Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings.}, language = {en} } @misc{KoltaiSchuette2017, author = {Koltai, P{\´e}ter and Sch{\"u}tte, Christof}, title = {A multi scale perturbation expansion approach for Markov state modeling of non-stationary molecular dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64868}, year = {2017}, abstract = {We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one.}, language = {en} } @misc{BittracherKoltaiKlusetal.2017, author = {Bittracher, Andreas and Koltai, P{\´e}ter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Sch{\"u}tte, Christof}, title = {Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63822}, year = {2017}, abstract = {We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.}, language = {en} } @misc{KoltaiCiccottiSchuette2016, author = {Koltai, Peter and Ciccotti, Giovanni and Sch{\"u}tte, Christof}, title = {On metastability and Markov state models for non-stationary molecular dynamics}, volume = {174103}, journal = {The Journal of Chemical Physics}, edition = {145}, issn = {1438-0064}, doi = {10.1063/1.4966157}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57869}, year = {2016}, abstract = {We utilize the theory of coherent sets to build Markov state models for non- equilibrium molecular dynamical systems. Unlike for systems in equilibrium, "meta- stable" sets in the non-equilibrium case may move as time evolves. We formalize this concept by relying on the theory of coherent sets, based on this we derive finite-time non-stationary Markov state models, and illustrate the concept and its main differences to equilibrium Markov state modeling on simple, one-dimensional examples.}, language = {en} } @misc{SarichSchuette2014, author = {Sarich, Marco and Sch{\"u}tte, Christof}, title = {Utilizing hitting times for finding metastable sets in non-reversible Markov chains}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-51209}, year = {2014}, abstract = {Techniques for finding metastable or almost invariant sets have been investigated, e.g., for deterministic dynamical systems in set-oriented numerics, for stochastic processes in molecular dynamics, and for random walks on complex networks. Most prominent algorithms are based on spectral apporaches and identify metastable sets via the doimant eigenvalues of the transfer operator associated with the dynamical system under consideration. These algorithms require the dominant eigenvalues to be real-valued. However, for many types of dynamics, e.g. for non-reversible Markov chains, this condition is not met. In this paper we utilize the hitting time apporach to metastable sets and demonstrate how the wellknown statements about optimal metastable decompositions of reversible chains can be reformulated for non-reversible chains if one switches from a spectral approach to an exit time approach. The performance of the resulting algorithm is illustrated by numerical experiments on random walks on complex networks.}, language = {en} } @misc{VegaSchuetteConrad2014, author = {Vega, Iliusi and Sch{\"u}tte, Christof and Conrad, Tim}, title = {SAIMeR: Self-adapted method for the identification of metastable states in real-world time series}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-50130}, year = {2014}, abstract = {In the framework of time series analysis with recurrence networks, we introduce SAIMeR, a heuristic self-adapted method that determines the elusive recurrence threshold and identifies metastable states in complex time series. To identify metastable states as well as the transitions between them, we use graph theory concepts and a fuzzy partitioning clustering algorithm. We illustrate SAIMeR by applying it to three real-world time series and show that it is able to identify metastable states in real-world data with noise and missing data points. Finally, we suggest a way to choose the embedding parameters used to construct the state space in which this method is performed, based on the analysis of how the values of these parameters affect two recurrence quantitative measurements: recurrence rate and entropy.}, language = {en} } @misc{LieFackeldeyWeber2013, author = {Lie, Han Cheng and Fackeldey, Konstantin and Weber, Marcus}, title = {A square root approximation of transition rates for a Markov State Model}, issn = {1438-0064}, doi = {10.1137/120899959}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42195}, year = {2013}, abstract = {Trajectory- or mesh-based methods for analyzing the dynamical behavior of large molecules tend to be impractical due to the curse of dimensionality - their computational cost increases exponentially with the size of the molecule. We propose a method to break the curse by a novel square root approximation of transition rates, Monte Carlo quadrature and a discretization approach based on solving linear programs. With randomly sampled points on the molecular energy landscape and randomly generated discretizations of the molecular configuration space as our initial data, we construct a matrix describing the transition rates between adjacent discretization regions. This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron cluster analysis and coarse-graining techniques in order to identify metastable sets in configuration space and approximate the transition rates between the metastable sets. Application of our method to a simple energy landscape on a two-dimensional configuration space provides proof of concept and an example for which we compare the performance of different discretizations. We show that the computational cost of our method grows only polynomially with the size of the molecule. However, finding discretizations of higher-dimensional configuration spaces in which metastable sets can be identified remains a challenge.}, language = {en} } @phdthesis{Roeblitz2009, author = {R{\"o}blitz, Susanna}, title = {Statistical Error Estimation and Grid-free Hierarchical Refinement in Conformation Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:188-fudissthesis000000008079-9}, year = {2009}, abstract = {The understanding of geometric structures and dynamical properties of molecular conformations gives insight into molecular long-term behavior. The identification of metastable conformations together with their life times and transition patterns is the intention of conformation dynamics. Conformation dynamics is a multi-scale approach that leads to a reduced description of the dynamical system in terms of a stochastic transition probability matrix. The present thesis deals with the error analysis of computed matrices and the resulting matrix functions. Since conformational membership vectors, as they are computed by the Robust Perron Cluster Analysis (PCCA+), form an invariant subspace of the transition matrix, subspace-based error estimators are of particular interest. The decomposition of the state space into basis functions and the approximation of integrals by Monte-Carlo quadrature give rise to row-wise correlated random matrices, for which stochastic norms are computed. Together with an appropriate statistical model for the distribution of matrix rows, this allows for the calculation of error bounds and error distributions of the invariant subspace and other variables of interest. Equilibration of errors among the basis functions can be achieved by enhanced sampling in regions where the trajectories are mixing slowly. Hierarchical refinement of such basis functions systematically improves the clustering into metastable conformations by reducing the error in the corresponding invariant subspace. These techniques allow for an evaluation of simulation results and pave the way for the analysis of larger molecules. Moreover, the extension of PCCA+ to non-reversible Markov chains, verified by the corresponding perturbation theory, and the modification of the objective function for the case of soft membership vectors represent a further generalization of the clustering method, thus continuing the development from PCCA over PCCA+ to PCCA++. The methods developed in this thesis are useful for but not limited to conformation dynamics. In fact, they are applicable to a broader class of problems which combine domain decomposition with Monte-Carlo quadrature. Possible application areas may include the chemical master equation or quantum dynamical systems.}, language = {en} } @misc{Weber2008, author = {Weber, Marcus}, title = {An efficient analysis of rare events in canonical ensemble dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10871}, number = {08-36}, year = {2008}, abstract = {For an analysis of a molecular system from a computational statistical thermodynamics point of view, extensive molecular dynamics simulations are very inefficient. During this procedure, at lot of redundant data is generated. Whereas the algorithms spend most of the computing time for a sampling of configurations within the basins of the potential energy landscape of the molecular system, the important information about the long-time behaviour of the molecules is given by transition regions and barriers between the basins, which are sampled rarely only. Thinking of molecular dynamics trajectories, researchers try to figure out which kind of dynamical model is suitable for an efficient simulation. This article suggests to change the point of view from extensive simulation of molecular dynamics trajectories to more efficient sampling strategies of the conformation dynamics approach.}, language = {en} } @misc{WeberKubeRiemeretal.2006, author = {Weber, Marcus and Kube, Susanna and Riemer, Alexander and Bujotzek, Alexander}, title = {Efficient Sampling of the Stationary Distribution of Metastable Dynamical Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9467}, number = {07-03}, year = {2006}, abstract = {In this article we aim at an efficient sampling of the stationary distribution of dynamical systems in the presence of metastabilities. In the past decade many sophisticated algorithms have been inven ted in this field. We do not want to simply add a further one. We address the problem that one has applied a sampling algorithm for a dynamical system many times. This leads to different samplings which more or less represent the stationary distribution partially very well, but which are still far away from ergodicity or from the global stationary distribution. We will show how these samplings can be joined together in order to get one global sampling of the stationary distribution.}, language = {en} } @misc{WeberWalterKubeetal.2006, author = {Weber, Marcus and Walter, Lionel and Kube, Susanna and Deuflhard, Peter}, title = {Stable computation of probability densities for metastable dynamical systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9331}, number = {06-39}, year = {2006}, abstract = {Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest to apply an aggregation/disaggregation method which only addresses wellconditioned sub-problems and thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples.}, language = {en} } @misc{Deuflhard2002, author = {Deuflhard, Peter}, title = {A Comparison of Related Concepts in Computational Chemistry and Mathematics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7138}, number = {02-46}, year = {2002}, abstract = {This article studies the relation of the two scientific languages Chemistry and Mathematics via three selected comparisons: (a) QSSA versus dynamic ILDM in reaction kinetics, (b) lumping versus discrete Galerkin methods in polymer chemistry, and (c) geometrical conformations versus metastable conformations in drug design. The common clear message from these comparisons is that chemical intuition may pave the way for mathematical concepts just as chemical concepts may gain from mathematical precisioning. Along this line, significant improvements in chemical research and engineering have already been possible -- and can be further expected in the future from the dialogue between the two scientific languages.}, language = {en} } @misc{CordesWeberSchmidtEhrenberg2002, author = {Cordes, Frank and Weber, Marcus and Schmidt-Ehrenberg, Johannes}, title = {Metastable Conformations via successive Perron-Cluster Cluster Analysis of dihedrals}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7074}, number = {02-40}, year = {2002}, abstract = {Decomposition of the high dimensional conformational space of bio-molecules into metastable subsets is used for data reduction of long molecular trajectories in order to facilitate chemical analysis and to improve convergence of simulations within these subsets. The metastability is identified by the Perron-cluster cluster analysis of a Markov process that generates the thermodynamic distribution. A necessary prerequisite of this analysis is the discretization of the conformational space. A combinatorial approach via discretization of each degree of freedom will end in the so called ''curse of dimension''. In the following paper we analyze Hybrid Monte Carlo simulations of small, drug-like biomolecules and focus on the dihedral degrees of freedom as indicators of conformational changes. To avoid the ''curse of dimension'', the projection of the underlying Markov operator on each dihedral is analyzed according to its metastability. In each decomposition step of a recursive procedure, those significant dihedrals, which indicate high metastability, are used for further decomposition. The procedure is introduced as part of a hierarchical protocol of simulations at different temperatures. The convergence of simulations within metastable subsets is used as an ''a posteriori'' criterion for a successful identification of metastability. All results are presented with the visualization program AmiraMol.}, language = {en} } @misc{FischerSchuetteDeuflhardetal.2001, author = {Fischer, Alexander and Sch{\"u}tte, Christof and Deuflhard, Peter and Cordes, Frank}, title = {Hierarchical Uncoupling-Coupling of Metastable Conformations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6296}, number = {01-03}, year = {2001}, abstract = {Uncoupling-coupling Monte Carlo (UCMC) combines uncoupling techniques for finite Markov chains with Markov chain Monte Carlo methodology. UCMC aims at avoiding the typical metastable or trapping behavior of Monte Carlo techniques. From the viewpoint of Monte Carlo, a slowly converging long-time Markov chain is replaced by a limited number of rapidly mixing short-time ones. Therefore, the state space of the chain has to be hierarchically decomposed into its metastable conformations. This is done by means of combining the technique of conformation analysis as recently introduced by the authors, and appropriate annealing strategies. We present a detailed examination of the uncoupling-coupling procedure which uncovers its theoretical background, and illustrates the hierarchical algorithmic approach. Furthermore, application of the UCMC algorithm to the \$n\$-pentane molecule allows us to discuss the effect of its crucial steps in a typical molecular scenario.}, language = {en} }